Average Error: 47.4 → 19.6
Time: 5.8m
Precision: 64
Internal Precision: 4160
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.6497181296145626 \cdot 10^{+103}:\\ \;\;\;\;\frac{2}{\left(\left(\left|\frac{k}{t}\right| \cdot t\right) \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)\right)\right) \cdot \left|\frac{k}{t}\right|}\\ \mathbf{if}\;t \le -9.849425412351248 \cdot 10^{-85}:\\ \;\;\;\;\frac{\frac{\frac{\frac{2}{t}}{t \cdot t}}{\sin k \cdot \tan k}}{\frac{\left|\frac{k}{t}\right|}{\ell} \cdot \frac{\left|\frac{k}{t}\right|}{\ell}}\\ \mathbf{if}\;t \le 1.7421167894681717 \cdot 10^{-264}:\\ \;\;\;\;\frac{2}{\left(\left(\left|\frac{k}{t}\right| \cdot t\right) \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)\right)\right) \cdot \left|\frac{k}{t}\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \sin k\right) \cdot \left|\frac{k}{t}\right|\right) \cdot \left|\frac{k}{t}\right|} \cdot \left(\frac{\ell}{t} \cdot \cos k\right)\\ \end{array}\]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Split input into 3 regimes

  2. if t < -1.6497181296145626e+103 or -9.849425412351248e-85 < t < 1.7421167894681717e-264

    1. Initial program 55.6

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt55.6

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\left(\sqrt{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \sqrt{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}\right)}}\]

    4. Applied simplify55.6

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{\left|\frac{k}{t}\right|} \cdot \sqrt{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}\right)}\]

    5. Applied simplify49.3

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left|\frac{k}{t}\right| \cdot \color{blue}{\left|\frac{k}{t}\right|}\right)}\]
    6. Using strategy rm

    7. Applied add-cube-cbrt49.3

      \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{{t}^{3}}\right) \cdot \sqrt[3]{{t}^{3}}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left|\frac{k}{t}\right| \cdot \left|\frac{k}{t}\right|\right)}\]

    8. Applied times-frac48.5

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{{t}^{3}}}{\ell} \cdot \frac{\sqrt[3]{{t}^{3}}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left|\frac{k}{t}\right| \cdot \left|\frac{k}{t}\right|\right)}\]

    9. Applied simplify48.5

      \[\leadsto \frac{2}{\left(\left(\left(\color{blue}{\frac{t}{\frac{\ell}{t}}} \cdot \frac{\sqrt[3]{{t}^{3}}}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left|\frac{k}{t}\right| \cdot \left|\frac{k}{t}\right|\right)}\]

    10. Applied simplify37.8

      \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\frac{\ell}{t}} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left|\frac{k}{t}\right| \cdot \left|\frac{k}{t}\right|\right)}\]
    11. Using strategy rm

    12. Applied associate-*r*31.3

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\left(\frac{t}{\frac{\ell}{t}} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left|\frac{k}{t}\right|\right) \cdot \left|\frac{k}{t}\right|}}\]
    13. Using strategy rm

    14. Applied *-un-lft-identity31.3

      \[\leadsto \frac{2}{\left(\left(\left(\left(\frac{t}{\frac{\ell}{t}} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left|\frac{k}{t}\right|\right) \cdot \color{blue}{\left(1 \cdot \left|\frac{k}{t}\right|\right)}}\]

    15. Applied associate-*r*31.3

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\left(\left(\frac{t}{\frac{\ell}{t}} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left|\frac{k}{t}\right|\right) \cdot 1\right) \cdot \left|\frac{k}{t}\right|}}\]

    16. Applied simplify24.1

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\left|\frac{k}{t}\right| \cdot t\right) \cdot \left(\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)\right)\right)} \cdot \left|\frac{k}{t}\right|}\]

    if -1.6497181296145626e+103 < t < -9.849425412351248e-85

    1. Initial program 29.8

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt29.8

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\left(\sqrt{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \sqrt{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}\right)}}\]

    4. Applied simplify29.7

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{\left|\frac{k}{t}\right|} \cdot \sqrt{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}\right)}\]

    5. Applied simplify22.5

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left|\frac{k}{t}\right| \cdot \color{blue}{\left|\frac{k}{t}\right|}\right)}\]

    6. Taylor expanded around -inf 63.0

      \[\leadsto \frac{2}{\left(\color{blue}{\frac{e^{3 \cdot \left(\log -1 - \log \left(\frac{-1}{t}\right)\right)} \cdot \sin k}{{\ell}^{2}}} \cdot \tan k\right) \cdot \left(\left|\frac{k}{t}\right| \cdot \left|\frac{k}{t}\right|\right)}\]

    7. Applied simplify10.2

      \[\leadsto \color{blue}{\frac{\frac{\frac{\frac{2}{t}}{t \cdot t}}{\sin k \cdot \tan k}}{\frac{\left|\frac{k}{t}\right|}{\ell} \cdot \frac{\left|\frac{k}{t}\right|}{\ell}}}\]

    if 1.7421167894681717e-264 < t

    1. Initial program 46.7

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt46.7

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \color{blue}{\left(\sqrt{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \sqrt{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}\right)}}\]

    4. Applied simplify46.7

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\color{blue}{\left|\frac{k}{t}\right|} \cdot \sqrt{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}\right)}\]

    5. Applied simplify39.0

      \[\leadsto \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left|\frac{k}{t}\right| \cdot \color{blue}{\left|\frac{k}{t}\right|}\right)}\]
    6. Using strategy rm

    7. Applied add-cube-cbrt39.1

      \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{\left(\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{{t}^{3}}\right) \cdot \sqrt[3]{{t}^{3}}}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left|\frac{k}{t}\right| \cdot \left|\frac{k}{t}\right|\right)}\]

    8. Applied times-frac37.0

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{{t}^{3}}}{\ell} \cdot \frac{\sqrt[3]{{t}^{3}}}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left|\frac{k}{t}\right| \cdot \left|\frac{k}{t}\right|\right)}\]

    9. Applied simplify36.9

      \[\leadsto \frac{2}{\left(\left(\left(\color{blue}{\frac{t}{\frac{\ell}{t}}} \cdot \frac{\sqrt[3]{{t}^{3}}}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left|\frac{k}{t}\right| \cdot \left|\frac{k}{t}\right|\right)}\]

    10. Applied simplify28.9

      \[\leadsto \frac{2}{\left(\left(\left(\frac{t}{\frac{\ell}{t}} \cdot \color{blue}{\frac{t}{\ell}}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left|\frac{k}{t}\right| \cdot \left|\frac{k}{t}\right|\right)}\]
    11. Using strategy rm

    12. Applied associate-*r*23.3

      \[\leadsto \frac{2}{\color{blue}{\left(\left(\left(\left(\frac{t}{\frac{\ell}{t}} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \tan k\right) \cdot \left|\frac{k}{t}\right|\right) \cdot \left|\frac{k}{t}\right|}}\]
    13. Using strategy rm

    14. Applied tan-quot23.3

      \[\leadsto \frac{2}{\left(\left(\left(\left(\frac{t}{\frac{\ell}{t}} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \color{blue}{\frac{\sin k}{\cos k}}\right) \cdot \left|\frac{k}{t}\right|\right) \cdot \left|\frac{k}{t}\right|}\]

    15. Applied associate-*l/23.3

      \[\leadsto \frac{2}{\left(\left(\left(\color{blue}{\frac{t \cdot \frac{t}{\ell}}{\frac{\ell}{t}}} \cdot \sin k\right) \cdot \frac{\sin k}{\cos k}\right) \cdot \left|\frac{k}{t}\right|\right) \cdot \left|\frac{k}{t}\right|}\]

    16. Applied associate-*l/22.6

      \[\leadsto \frac{2}{\left(\left(\color{blue}{\frac{\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k}{\frac{\ell}{t}}} \cdot \frac{\sin k}{\cos k}\right) \cdot \left|\frac{k}{t}\right|\right) \cdot \left|\frac{k}{t}\right|}\]

    17. Applied frac-times22.1

      \[\leadsto \frac{2}{\left(\color{blue}{\frac{\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \sin k}{\frac{\ell}{t} \cdot \cos k}} \cdot \left|\frac{k}{t}\right|\right) \cdot \left|\frac{k}{t}\right|}\]

    18. Applied associate-*l/18.4

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \sin k\right) \cdot \left|\frac{k}{t}\right|}{\frac{\ell}{t} \cdot \cos k}} \cdot \left|\frac{k}{t}\right|}\]

    19. Applied associate-*l/19.3

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(\left(\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \sin k\right) \cdot \left|\frac{k}{t}\right|\right) \cdot \left|\frac{k}{t}\right|}{\frac{\ell}{t} \cdot \cos k}}}\]

    20. Applied associate-/r/19.2

      \[\leadsto \color{blue}{\frac{2}{\left(\left(\left(\left(t \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \sin k\right) \cdot \left|\frac{k}{t}\right|\right) \cdot \left|\frac{k}{t}\right|} \cdot \left(\frac{\ell}{t} \cdot \cos k\right)}\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 5.8m)Debug logProfile

herbie shell --seed '#(1071979731 1496239409 439705970 2863295848 982327776 189749553)' 
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10-)"
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (- (+ 1 (pow (/ k t) 2)) 1))))